| L(s) = 1 | − 2·3-s + 5-s − 4·7-s + 9-s + 2·11-s + 13-s − 2·15-s + 2·17-s − 6·19-s + 8·21-s + 6·23-s + 25-s + 4·27-s − 2·29-s − 6·31-s − 4·33-s − 4·35-s + 2·37-s − 2·39-s + 10·41-s + 10·43-s + 45-s − 12·47-s + 9·49-s − 4·51-s − 2·53-s + 2·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 0.516·15-s + 0.485·17-s − 1.37·19-s + 1.74·21-s + 1.25·23-s + 1/5·25-s + 0.769·27-s − 0.371·29-s − 1.07·31-s − 0.696·33-s − 0.676·35-s + 0.328·37-s − 0.320·39-s + 1.56·41-s + 1.52·43-s + 0.149·45-s − 1.75·47-s + 9/7·49-s − 0.560·51-s − 0.274·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.998195685405704786191863974410, −6.88790761852876821998572696853, −6.50582599523088593057290597481, −5.93751448677235778004080810798, −5.31195556198861378291649819015, −4.29032007651445818827415518174, −3.43852159309911188911327625151, −2.50577777165118673567864114027, −1.11941535085904868961137140184, 0,
1.11941535085904868961137140184, 2.50577777165118673567864114027, 3.43852159309911188911327625151, 4.29032007651445818827415518174, 5.31195556198861378291649819015, 5.93751448677235778004080810798, 6.50582599523088593057290597481, 6.88790761852876821998572696853, 7.998195685405704786191863974410
